A note on the sum of cubic residues with even and odd index

V. P. Ramesh (), Gowtham R. () and Saswati Sinha ()
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V. P. Ramesh: Central University of Tamil Nadu
Gowtham R.: Indian Institute of Technology Madras
Saswati Sinha: Central University of Tamil Nadu

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 1, 388-391

Abstract: Abstract Let p be an odd prime and $$\ell \in {\mathbb {N}}$$ ℓ ∈ N . We prove that the sum of cubic residues of $$p^\ell $$ p ℓ with even index equals the sum of cubic residues of $$p^\ell $$ p ℓ with odd index if and only if $$p \equiv 1 \pmod 4$$ p ≡ 1 ( mod 4 ) . We also prove that for $$2p^\ell $$ 2 p ℓ , $$p \equiv 1 \pmod 4$$ p ≡ 1 ( mod 4 ) is sufficient but not necessary for the two sums to be equal. We also present a closed-form expression for the sum and study some properties.

Keywords: Cubic residues; Primitive roots; Indices; 11A07; 11A41 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00370-w

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